2,555 research outputs found
Minimal generating sets of non-modular invariant rings of finite groups
It is a classical problem to compute a minimal set of invariant polynomial
generating the invariant ring of a finite group as an algebra. We present here
an algorithm for the computation of minimal generating sets in the non-modular
case. Apart from very few explicit computations of Groebner bases, the
algorithm only involves very basic operations, and is thus rather fast.
As a test bed for comparative benchmarks, we use transitive permutation
groups on 7 and 8 variables. In most examples, our algorithm implemented in
Singular works much faster than the one used in Magma, namely by factors
between 50 and 1000. We also compute some further examples on more than 8
variables, including a minimal generating set for the natural action of the
cyclic group of order 11 in characteristic 0 and of order 15 in characteristic
2.
We also apply our algorithm to the computation of irreducible secondary
invariants.Comment: 14 pages v3: Timings updated. One example adde
Effective Invariant Theory of Permutation Groups using Representation Theory
Using the theory of representations of the symmetric group, we propose an
algorithm to compute the invariant ring of a permutation group. Our approach
have the goal to reduce the amount of linear algebra computations and exploit a
thinner combinatorial description of the invariant ring.Comment: Draft version, the corrected full version is available at
http://www.springer.com
Some relational structures with polynomial growth and their associated algebras II: Finite generation
The profile of a relational structure is the function which
counts for every integer the number, possibly infinite, of
substructures of induced on the -element subsets, isomorphic
substructures being identified. If takes only finite values, this
is the Hilbert function of a graded algebra associated with , the age
algebra , introduced by P.~J.~Cameron.
In a previous paper, we studied the relationship between the properties of a
relational structure and those of their algebra, particularly when the
relational structure admits a finite monomorphic decomposition. This
setting still encompasses well-studied graded commutative algebras like
invariant rings of finite permutation groups, or the rings of quasi-symmetric
polynomials.
In this paper, we investigate how far the well know algebraic properties of
those rings extend to age algebras. The main result is a combinatorial
characterization of when the age algebra is finitely generated. In the special
case of tournaments, we show that the age algebra is finitely generated if and
only if the profile is bounded. We explore the Cohen-Macaulay property in the
special case of invariants of permutation groupoids. Finally, we exhibit
sufficient conditions on the relational structure that make naturally the age
algebra into a Hopf algebra.Comment: 27 pages; submitte
Operator bases, -matrices, and their partition functions
Relativistic quantum systems that admit scattering experiments are
quantitatively described by effective field theories, where -matrix
kinematics and symmetry considerations are encoded in the operator spectrum of
the EFT. In this paper we use the -matrix to derive the structure of the EFT
operator basis, providing complementary descriptions in (i) position space
utilizing the conformal algebra and cohomology and (ii) momentum space via an
algebraic formulation in terms of a ring of momenta with kinematics implemented
as an ideal. These frameworks systematically handle redundancies associated
with equations of motion (on-shell) and integration by parts (momentum
conservation).
We introduce a partition function, termed the Hilbert series, to enumerate
the operator basis--correspondingly, the -matrix--and derive a matrix
integral expression to compute the Hilbert series. The expression is general,
easily applied in any spacetime dimension, with arbitrary field content and
(linearly realized) symmetries.
In addition to counting, we discuss construction of the basis. Simple
algorithms follow from the algebraic formulation in momentum space. We
explicitly compute the basis for operators involving up to scalar fields.
This construction universally applies to fields with spin, since the operator
basis for scalars encodes the momentum dependence of -point amplitudes.
We discuss in detail the operator basis for non-linearly realized symmetries.
In the presence of massless particles, there is freedom to impose additional
structure on the -matrix in the form of soft limits. The most na\"ive
implementation for massless scalars leads to the operator basis for pions,
which we confirm using the standard CCWZ formulation for non-linear
realizations.Comment: 75 pages plus appendice
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